Office Applications and Entertainment, Magic Cubes Index About the Author

 7.0    Special Cubes, Prime Numbers 7.11   Magic Cubes, Sum of Latin Cubes (2) 7.11.1 Introduction As illustrated in Attachment 7.10.3, order 8 Composed Magic Squares can be based on the four horizontal Magic Planes of order 4 Magic Cubes. Consequently Prime Number Magic Cubes might be constructed based on two suitable selected Latin Cubes, Comparable with order 8 Prime Number Magic Squares (ref. Section 14.12.5). 7.11.2 Simple Magic Cubes (4 x 4 x 4) The elements of two Latin Cubes A and B, with latin space diagoanls, might result in a Simple Prime Number Magic Cube C with elements ci = ai + bi, i = 1 ... 64 as illustrated below for an Associated Magic Cube:
A
 1583 1019 3793 3229 19 3793 1019 4793 3229 83 4729 1583 4793 4729 83 19
B
 504 0 690 1194 480 714 84 1110 714 480 1110 84 690 1194 504 0
MC = 12012
 2087 1019 4483 4423 499 4507 1103 5903 3943 563 5839 1667 5483 5923 587 19

 19 83 4729 4793 1583 4729 83 3229 4793 1019 3793 19 3229 3793 1019 1583

 690 1194 504 0 714 480 1110 84 480 714 84 1110 504 0 690 1194

 709 1277 5233 4793 2297 5209 1193 3313 5273 1733 3877 1129 3733 3793 1709 2777

 3229 3793 1019 1583 4793 1019 3793 19 1583 4729 83 3229 19 83 4729 4793

 0 504 1194 690 84 1110 480 714 1110 84 714 480 1194 690 0 504

 3229 4297 2213 2273 4877 2129 4273 733 2693 4813 797 3709 1213 773 4729 5297

 4793 4729 83 19 3229 83 4729 1583 19 3793 1019 4793 1583 1019 3793 3229

 1194 690 0 504 1110 84 714 480 84 1110 480 714 0 504 1194 690

 5987 5419 83 523 4339 167 5443 2063 103 4903 1499 5507 1583 1523 4987 3919
 The key to possible solutions is to find Correlated Balanced Magic Lines {ai, i = 1 ... 8} and {bj, j = 1 ... 8} such that cij = ai + bj (i,j = 1 ... 8) are distinct prime numbers (64 ea). A few of such sets of Correlated Balanced Magic Lines are shown in following table:
 a1 a2 a3 a4 a5 a6 a7 a8 - b1 b2 b3 b4 b5 b6 b7 b8 - Sa Sb S8 19 83 1019 1583 3229 3793 4729 4793 - 0 84 480 504 690 714 1110 1194 - 19248 4776 24024 61 281 661 947 1201 1487 1867 2087 - 0 1632 1716 2052 2310 2646 2730 4362 - 8592 17448 26040 229 463 859 1153 6469 6763 7159 7393 - 0 448 724 1230 2068 2574 2850 3298 - 30488 13192 43680 197 257 457 2797 3467 5807 6007 6067 - 0 204 1170 2202 2622 3654 4620 4824 - 25056 19296 44352 241 503 773 3413 5101 7741 8011 8273 - 0 588 678 726 1890 1938 2028 2616 - 34056 10464 44520 241 331 631 1453 4201 5023 5323 5413 - 0 798 1240 1548 4018 4326 4768 5566 - 22616 22264 44880
 Attachment 7.11.2 shows for each set of Correlated Magic Lines shown above, an example of the resulting Prime Number Associated Magic Cubes (MC = s8/2). 7.11.3 Associated Latin Cubes (4 x 4 x 4) Associated Latin Cubes, can be obtained by applying the linear equations shown in Section 7.2.3. For Latin Cubes based on the integers {0, 1 ... 7} the related Magic Sum s1 = 28/2 = 14. Based on the equations mentioned above, a fast routine can be written to generate the defined Associated Latin Cubes of order 4 (ref. LtnCbs4d1). Subject routine produced, with a(64) = 0 and a(63) = 1 ... 3, 12240 Associated Latin Cubes within 267 seconds. Prime Number Associated Magic Cubes C can be generated by selecting combinations of Latin Cubes (A, B) while: substituting the integers {0, 1 ... 7} by {ai, i = 1 ... 8} and {bj, j = 1 ... 8}; ensuring that for the resulting cube C the numbers cij = ai + bj (i,j = 1 ... 8) are distinct prime numbers (64 ea). which can be achieved with routine CnstrCbs4b. Due to the vast amount of Associated Latin Cubes, a more controllable collection of Associated Magic Cubes can be obtained by means of Self Orthogonal Latin Cubes. The principle is illustrated below for the equivalent order 8 Latin Squares of the Associated Latin - and resulting Magic Cubes shown in Section 7.11.2 above.
A8
 1583 1019 3793 3229 19 83 4729 4793 19 3793 1019 4793 1583 4729 83 3229 3229 83 4729 1583 4793 1019 3793 19 4793 4729 83 19 3229 3793 1019 1583 3229 3793 1019 1583 4793 4729 83 19 4793 1019 3793 19 3229 83 4729 1583 1583 4729 83 3229 19 3793 1019 4793 19 83 4729 4793 1583 1019 3793 3229
B8 = T(A8)
 504 0 690 1194 690 1194 504 0 480 714 84 1110 714 480 1110 84 714 480 1110 84 480 714 84 1110 690 1194 504 0 504 0 690 1194 0 504 1194 690 1194 690 0 504 84 1110 480 714 1110 84 714 480 1110 84 714 480 84 1110 480 714 1194 690 0 504 0 504 1194 690
C8 = A8 + B8
 2087 1019 4483 4423 709 1277 5233 4793 499 4507 1103 5903 2297 5209 1193 3313 3943 563 5839 1667 5273 1733 3877 1129 5483 5923 587 19 3733 3793 1709 2777 3229 4297 2213 2273 5987 5419 83 523 4877 2129 4273 733 4339 167 5443 2063 2693 4813 797 3709 103 4903 1499 5507 1213 773 4729 5297 1583 1523 4987 3919
 Square A8 is Semi-Latin with Latin Rows and Latin Diagonals (Symmetrical). The Semi-Latin Square B8 is the transposed square of A8 (rows and columns exchanged). Based on this principle, a routine can be written to generate Prime Number Associated Magic Cubes of order 4 (ref. LtnCbs4d2). Subject routine counted 2304 Prime Number Assocociated Magic Cubes within an hour (MC = 12012), of wich 32 are shown in Attachment 7.11.3. Attachment 7.11.2 shows for each set of Correlated Magic Lines shown in Section 7.10.2 above, an example of the resulting Prime Number Associated Magic Cubes. 7.11.4 Associated Latin Cubes (4 x 4 x 4)        3D Compact Associated, 3D Compact Latin Cubes can be obtained by the linear equations shown in Section 7.2.4. For Latin Cubes based on the integers {0, 1 ... 7} the related Magic Sum s1 = 28/2 = 14. Based on the equations mentioned above, a fast routine can be written to generate the defined Associated Latin Cubes of order 4 (ref. LtnCbs44). Subject routine produced 6528 (= 17 * 384) Associated, 3D Compact Latin Cubes within 45 seconds. Based on this collection, 2.654.208 Associated, 3D Compact Magic Cubes can be constructed (ref. CnstrCbs4b). The graph shown above, shows the frequency n4 of subject Associated, 3D Compact Magic Cubes as a function of n (multiples of 384). Attachment 7.11.4 shows for each set of Correlated Magic Lines shown in Section 7.10.2 above, an example of the resulting Prime Number Associated, 3D Compact Magic Cubes. 7.11.5 Simple Latin Cubes (4 x 4 x 4)        Horizontal Associated Magic Planes Simple Latin Cubes with Horizontal Associated Planes can be obtained by applying the linear equations shown in Section 7.2.2. For Latin Cubes based on the integers {0, 1 ... 7} the related Magic Sum s1 = 28/2 = 14. Based on the equations mentioned above, a fast routine can be written to generate the defined Simple Latin Cubes of order 4 (ref. LtnCbs41). Subject routine produced 768 Simple Latin Cubes with Horizontal Associated Magic Planes within 17 seconds. Based on this collection, 294912 Magic Cubes with Horizontal Associated Magic Planes can be constructed (ref. CnstrCbs4b). Attachment 7.11.5 shows for each set of Correlated Magic Lines shown in Section 7.10.2 above, an example of the resulting Prime Number Simple Magic Cubes with Horizontal Associated Magic Planes. 7.11.6 Simple Latin Cubes (4 x 4 x 4)        Horizontal Pan Magic Planes Simple Latin Cubes with Horizontal Pan Magic Planes can be obtained by applying the linear equations shown in Section 7.2.1. For Latin Cubes based on the integers {0, 1 ... 7} the related Magic Sum s1 = 28/2 = 14. Based on the equations mentioned above, a fast routine can be written to generate the defined Simple Latin Cubes of order 4 (ref. LtnCbs42). Subject routine produced 1536 Simple Latin Cubes with Horizontal Pan Magic Planes within 7,7 seconds. Based on this collection, 589824 Magic Cubes with Horizontal Pan Magic Planes can be constructed (ref. CnstrCbs4b). Attachment 7.11.6 shows for each set of Correlated Magic Lines shown in Section 7.10.2 above, an example of the resulting Prime Number Simple Magic Cubes with Horizontal Pan Magic Planes. 7.11.7 Simple Latin Cubes (4 x 4 x 4)        Pantriagonal and Complete Pantriagonal and Complete (Semi) Latin Cubes can be obtained by applying the linear equations shown in Section 7.2.6. For (Semi) Latin Cubes based on the integers {0, 1 ... 7} the related Magic Sum s1 = 28/2 = 14. Based on the equations mentioned above, a fast routine can be written to generate the defined Simple (Semi) Latin Cubes of order 4 (ref. LtnCbs46). Subject routine produced 53760 Pantriagonal and Complete (Semi) Latin Cubes within 529 seconds. Based on the sub collection of the first 1268 (Semi) Latin Cubes, 1440 Pantriagonal and Complete Magic Cubes can be constructed (ref. CnstrCbs4b). Attachment 7.11.7 shows for each set of Correlated Magic Lines shown in Section 7.10.2 above, an example of the resulting Prime Number Pantriagonal and Complete Magic Cubes. 7.11.8 Summary The obtained results regarding the miscellaneous Prime Number Magic Cubes as deducted and discussed in previous sections are summarized in following table:
 Type Main Characteristics Subroutine Results - - - - Latin Cubes Associated > 12240 Associated, 3D Compact 6528 Simple, Horizontal Associated Planes 192 Simple, Horizontal Pan Magic Planes 1536 Pantriagonal and Complete 53760 Magic Cubes Associated, MC = 12012 Associated Associated,  3D Compact Simple, Hor. Associated Magic Planes Simple, Hor. Pan Magic Planes Pantriagonal and Complete - - - -
 This is the end of the Chapter 'Prime Number Magic Cubes' of this website.